Domain 1 Lesson 4 Absolute Value Common Core Standards: 6.NS.7.c, 6.NS.7.d Getting the Idea The absolute value of a number is its distance from 0 on a number line. Since a distance must be either a positive number or zero, the absolute value of a number is always a positive number or zero. The absolute value of a number x is written as x. The integers 24 and 4 are opposites. You can use the number line below to see that each number is the same distance from 0. So, 24 5 4 and 4 5 4. 8 7 6 5 4 3 2 1 0 1 2 3 4 5 6 7 8 opposites Example 1 Find the value of 27. Strategy Step 1 Use a number line. Plot a point for 27 on a number line. 8 7 6 5 4 3 2 1 0 1 2 3 4 5 6 7 8 Step 2 Count the number of units from 27 to 0. 7 units 8 7 6 5 4 3 2 1 0 1 2 3 4 5 6 7 8 Solution 27 5 7 The distance is 7 units. 27 5 7 39
You can use absolute values to represent and help you understand real-world situations. For example, if a diver is 20 meters below the ocean s surface, that depth, in meters, can be shown as 220 meters. But the distance the diver would have to swim to get to the surface of the water cannot be represented by a negative number. You can use absolute value instead. The diver must swim 220 meters, or 20 meters, to reach the surface. Example 2 Hannah wrote a check for more money than she has in her bank account. The balance in her account is now 2$60. How much does Hannah owe the bank, in dollars? Strategy Step 1 Step 2 Use an absolute value to represent the situation. Is the amount she owes a positive or negative number? The balance in Hannah s account is 2$60, but she cannot owe the bank a negative amount of money. The amount Hannah owes must be shown as a positive number. Use an absolute value. The amount she owes, in dollars, is 260, or 60. The number line below shows that Hannah owes the bank $60. 60 units 80 70 60 50 40 30 20 10 0 10 20 30 40 50 60 70 80 Solution Hannah owes the bank $60. 40 Domain 1: The Number System
Lesson 4: Absolute Value Absolute values can also help you understand situations in which an exact number is not known. Example 3 A team of mountaineers has climbed to the summit of Mount Everest. The temperature at the summit is less than 2158F. Describe how many degrees Fahrenheit below 08F the temperature is. Strategy Step 1 Step 2 Use an absolute value to represent and understand the situation. Is the number of degrees below 08F a positive or negative number? An actual temperature may be negative, but the number of degrees Fahrenheit below 08F must be a positive number. Use a number line to represent the situation. The temperature is less than 2158F. On a number line, a number less than 215 is to the left of 215. The arrow below shows all the numbers less than 215. less than 15 30 25 20 15 10 5 0 5 10 15 20 25 30 Step 3 Use absolute value to describe the number of degrees Fahrenheit below 08F. 215 5 15 less than 15 15 units 30 25 20 15 10 5 0 5 10 15 20 25 30 All the numbers less than 215 are more than 15 units from 0. So, if the temperature is less than 2158F, it is more than 158F below 08F. Solution The temperature at the summit is more than 158F below 08F. 41
Coached Example Yesterday, Marcus bought two different stocks, A and B, each at the same price. From yesterday to today, the change in the price of Stock A was 2$12, and the change in the price of Stock B was $9. From yesterday to today, which stock s price changed by the greatest amount? The price change with the greatest is the greatest change. On the number line below, plot points for 212 and 9. 14 13 12 11 10 9 8 7 6 5 4 3 2 1 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 Count the units from each integer to 0 to determine its absolute value. 212 5 9 5 Which number has the greater absolute value, 212 or 9? The stock with the price change of dollars changed by the greatest amount. That stock was Stock. 42 Domain 1: The Number System
Lesson 4: Absolute Value Lesson Practice Choose the correct answer. 1. Which point represents a number with an absolute value of 1? A B C D 5 4 3 2 1 0 1 2 3 4 5 A. point A B. point B C. point C D. point D 2. Which statement best describes 272? A. the distance from 272 to 72 on a number line B. the distance from 27 to 22 on a number line C. the distance from 27 to 2 on a number line D. the distance from 272 to 0 on a number line 4. Which point or points on the number line represent numbers with absolute values of 4? J K L M N 5 4 3 2 1 0 1 2 3 4 5 A. points J and N B. points K and M C. points K and N D. points L and N 5. Lorraine s checking account has a balance of less than 2$200. Which statement is true about how much Lorraine owes the bank? A. Lorraine owes exactly $200. B. Lorraine owes exactly 2$200. C. Lorraine owes less than $200. D. Lorraine owes more than $200. 3. A scientist stores liquid nitrogen at a temperature of 23318F. Exactly how many degrees Fahrenheit below 08F is the liquid nitrogen? A. 331 B. 31 C. 0 D. 2331 43
Use the information for questions 6 and 7. Max is a diver. He uses positive numbers to represent elevations above the water s surface and negative numbers to represent elevations below the water s surface. Max is standing on a springboard. He represents his location as 3 meters. He lets a ring drop to the bottom of the pool. He represents its location at the bottom of the pool as 24 meters. 6. If Max dives into the pool from the springboard, how many meters will he fall vertically before he reaches the water s surface? A. 4 meters C. 23 meters B. 3 meters D. 24 meters 7. How many meters below the surface of the water is the ring? A. more than 4 meters B. less than 4 meters C. exactly 4 meters D. exactly 3 meters 8. A football coach recorded the results of his team s first 4 plays in its last game. The table below shows his data. Football Plays Play Number of Yards 1 8 2 22 3 5 4 27 A. During which play did the team lose the fewest yards? Use what you know about absolute value and the number line below to explain how you determined your answer. 8 7 6 5 4 3 2 1 0 1 2 3 4 5 6 7 8 B. During which play did the team gain the most yards? Use what you know about absolute value and the number line above to explain your answer. 44 Domain 1: The Number System
Lesson 4: Absolute Value 9. Select every point that has an absolute value of 2. A B C D E 5 4 3 2 1 0 1 2 3 4 5 A. Point A B. Point B C. Point C D. Point D E. Point E 10. Select True or False for each statement. A. 258 is equal to the distance from 258 to 58 on a number line. True False B. 258 is equal to the distance from 0 to 58 on a number line. True False C. 258 is equal to the distance from 0 to 85 on a number line. True False D. 258 is equal to the distance from 258 to 0 on a number line. True False 11. Walter s checking account has a balance of 2$32. How much does Walter owe his bank, in dollars? 32 Walter owes the bank 0 32. 45
12. Compare the location of each point to the absolute value of 3. Write the letter for each point in the correct box. S T U V W X 5 4 3 2 1 0 1 2 3 4 5 S T U V W X Absolute Value Equal to 3 Absolute Value Not Equal to 3 13. Use a number from the box to complete each sentence. A B C D 5 4 3 2 1 0 1 2 3 4 5 4 Point A has an absolute value of. Point B has an absolute value of. Point C has an absolute value of. Point D has an absolute value of. 3 3 4 14. Jane is rock climbing. Before lunch, her elevation is represented by 15 feet. After lunch, her elevation is represented by 24 feet. Look at each statement. Is the statement true? Select Yes or No. A. Jane is 4 feet above sea level after lunch. Yes No B. Jane is 4 feet below sea level before lunch. Yes No C. Jane is 4 feet below sea level after lunch. Yes No D. Jane is 15 feet above sea level before lunch. Yes No E. Jane is 15 feet above sea level after lunch. Yes No F. Jane is 15 feet below sea level before lunch. Yes No 46 Domain 1: The Number System